Re: [問題] 請問如何從incompressible導出另一定義
※ 引述《Lanjaja ()》之銘言：
: 【出處】(習題或問題的出處)
: 看書和上網查有的問題。
: 【題目】(題目的文字敘述，如有圖片亦可提供圖片)
: 一般認知的incompressible定義是div(V) = 0
: 可是我在書上和wiki都有提到一種定義是(1/ρ)@ρ/@P ~ 0，
: 只是提及，未有證明兩種定義是否有何關係，如何互推。
: 【瓶頸】(解題瓶頸或思考脈絡，請盡量詳述以利回答者知道要從何處講解指導)
: 由連續方程式
: @ρ/@t + div(ρV) = 0
: 得到Dρ/Dt + ρdiv(V) = 0
: D是物質微分
: 因為incompressible的條件
: 推到Dρ/Dt = 0
: 再得出@ρ/@t + (@ρ/@P)(DP/Dt) = 0
: 接下來就做不出來了...
: 可以請強者幫忙救援一下嗎？
: 感謝回答
First of all, we need to know in general, the compressibility itself is a quan
tity that depends on the process. So basically, they are different concepts, d
iv(v)=0 tells you the fluid is incompressible dynamically, (∂ρ/∂P)_T=0 tell
s you the fluid is incomprehensible under isothermic process. If the fluid is
flowing isothermically, then they are equivalent in this case, otherwise they
may be different.
From mass conservation, we know dρ/dt=0 is equivalent to div(v)=0. so the que
stion is how to relate dρ/dt to (∂ρ/∂P).
As ρ is a thermodynamic intensive quantity, for ordinary fluid, it only has 2
degrees of freedom to control, so if the fluid flows without changes of tempe
rature, dρ/dt= (∂ρ/∂P)_T dP/dt, as we know the pressure of a volume elemen
t will change as the fluid flows, dρ/dt=0 is equivalent to (∂ρ/∂P)_T=0.
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 47.20.59.170 (美國)
※ 文章網址: https://www.ptt.cc/bbs/Physics/M.1640188940.A.E26.html
推
12/23 01:23,
2年前
, 1F
12/23 01:23, 1F
→
12/23 01:24,
2年前
, 2F
12/23 01:24, 2F
討論串 (同標題文章)
本文引述了以下文章的的內容：
完整討論串 (本文為第 2 之 2 篇)：